Fast Simulation of Cellular Networks with Dynamic …vazquez/papers/Tomacs01.pdfFast Simulation of Cellular Networks with Dynamic Channel Assignment Felisa J. Vazquez-Abad´ Department

Fast Simulation of Cellular Networks with DynamicChannel Assignment

Felisa J. Vazquez-Abad�

Department of Computer Science and Operations Research

University of Montreal, Montreal, Canada H3C 3J7

Email: [email protected]

also Principal Fellow, Department of Electrical and Electronic Engineering

The University of Melbourne

Lachlan L. H. Andrew and David Everitt�

ARC Special Research Centre for Ultra-Broadband Information Networks

Department of Electrical and Electronic Engineering

The University of Melbourne, Victoria 3010, Australia

Email: � l.andrew,d.everitt � @ee.mu.oz.au

Abstract

Blocking probabilities in cellular mobile communication networks using dynamic channelassignment are hard to compute for realistic sized systems. This computational difficulty is dueto the structure of the state space, which imposes strong coupling constraints amongst compo-nents of the occupancy vector. Approximate tractable models have been proposed, which haveproduct form stationary state distributions. However, for real channel assignment schemes,the product form is a poor approximation and it is necessary to simulate the actual occupancyprocess in order to estimate the blocking probabilities.

Meaningful estimates of the blocking probability typically require an enormous amountof CPU time for simulation, since blocking events are usually rare. Advanced simulationapproaches use importance sampling (IS) to overcome this problem. In this paper we study tworegimes under which blocking is a rare event: low load and high cell capacity. Our simulationsuse the standard clock (SC) method. For low load, we propose a change of measure that wecall static ISSC, which has bounded relative error. For high capacity, we use a change ofmeasure that depends on the current state of the network occupancy. This is the dynamic ISSCmethod. We prove that this method yields zero variance estimators for single clique models,and we empirically show the advantages of this method over naıve simulation for networks ofmoderate size and traffic loads.�

Supported in part by NSERC-Canada grant # WFA0184198.�Supported by the Australian Research Council (ARC).

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1 Introduction

Efficient design of cellular mobile communications networks requires the ability to determine thequality of service provided by a particular network configuration. A common quality of servicemeasure is the blocking probability, which is the probability that a new call will not be admittedto the network due to insufficient network resources. This paper will consider techniques fordetermining the blocking probability in cellular telephony systems with frequency reuse, includingfirst generation systems such as the Advanced Mobile Phone System, AMPS [Lee, 1995], andsecond generation systems such as the Global System for Mobile communication, GSM [Moulyand Pautet, 1992, Redl, Weber, and Oliphant, 1995].

In cellular networks, each mobile station communicates with a base station connected to thewireline telephone network. The region in which mobiles connect to a given station is called acell. Each mobile station communicates with its base station using a specific frequency pair orfrequency/time-slot pair known as a “channel”. To avoid interference, this channel cannot be usedin nearby cells; however, it may be reused in cells sufficiently remote that interference caused bythe reused channel is below a specified threshold.

In static assignment schemes, each cell is allocated a fixed subset of the available channels, andcalls arriving in a cell are connected only when there are free channels available from that subset.While simple to implement, this strategy may result in wasted resources; all the channels for onecell may be in use, but adjacent cells may have free capacity that could be used to connect incom-ing calls without causing interference. Network capacity can be improved by dynamic channelassignment [Cox and Reudink, 1972, Cox and Reudink, 1973], in which channels not currently inuse in the nearby cells may be used. It is these systems which are the focus of this paper.

Many techniques have been developed for determining the performance of such networks. ForMarkov models (Poisson arrivals and exponential holding times), when the system is reversible[Kelly, 1979], the stationary state distribution has a simple product form expression on a statespace � which is a small subset of a hypercube � . When there is no mobility of users, there issuch a product form solution if the network uses “maximum packing” [Everitt and Macfadyen,1983], in which calls in progress can be rearranged to use different channels. There are also mod-els of mobility which preserve this property (see, for example, Pallant and Taylor, 1995, Boucherieand Mandjes, 1998). Moreover, the result remains valid even when call holding times have non-exponential distributions [Kelly, 1979]. Product form systems have been studied extensively (see,for example, the survey of Nelson, 1993). The product form expression involves a normalizingconstant, from which the blocking probability can be determined directly, without needing to de-termine specific state probabilities. It can be evaluated exactly by recursive methods [Dziong andRoberts, 1987, Pinsky and Conway, 1992], mean value analysis [Reiser and Lavenberg, 1980],generating function inversion methods [Choudhury, Leung, and Whitt, 1995] or uniform asymp-totic approximation [Mitra and Morrison, 1994]. However, these techniques all have exponentialcomplexity in the number of cells.

For systems with a large number of cells, Monte Carlo techniques can be used either to estimatethe normalizing constant [Ross and Wang, 1992] or to estimate blocking in a way that avoids theneed to calculate it. The simplest approach of the second type is the acceptance/rejection (A/R)

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method, in which states are generated in the full hypercube � ; those lying outside the state spaceare rejected, while for those on the boundary of the feasible region, the proportion of blocked

cells is recorded, weighted by the respective arrival rates (see, for example, Everitt and Macfadyen,1983). As the number of cells grows, generation of a sample point inside the state space

� �may become a rare event, and so importance sampling (IS) has been applied to these methods(see Ross, Tsang, and Wang, 1994, Mandjes, 1997, Lassila and Virtamo, 2000). An alternativeapproach is to use Markov Chain Monte Carlo (MCMC) techniques such as the Gibbs samplerused by Lassila and Virtamo [1998] and Vazquez-Abad and Andrew [2000]. These generate aMarkov chain whose steady state probabilities satisfy the target product form, and they may besimulated more efficiently.

Most dynamic channel assignment implementations do not have such a product form solution.It is common in such cases to use closed form approximations, such as the ubiquitous reducedload approximation, developed for circuit switched networks (see, for example, Kelly, 1991). Thisapproximation works well if there is minimal correlation between blocking due to conflicts withdifferent reuse constraints, but poorly if there is significant correlation. Due to the spatial nature ofthe reuse constraints in the cellular case, it can be expected that there will be significant correlation.There are many other approximations; see, for example, Zahorjan, Eager, and Sweillam [1988].

A very flexible, straightforward and hence common approach is to directly simulate the arrivaland departure process of calls. This allows any performance measure of the system to be estimated.Moreover, it allows arbitrary channel allocation schemes to be compared, including those for whichthere is no product form solution, or indeed no known closed form solution at all. For thesereasons, this is the approach most commonly taken by engineers investigating different dynamicchannel assignment systems. However, this approach can be very slow, especially when blockingprobabilities are low. In this paper, we present two importance sampling schemes for the efficientsimulation of systems with low blocking probabilities, assuming no user mobility.

Section 2 outlines some important background material, starting with the model for the channeloccupancy process, and then describes the principles of fast simulation. Section 3 describes theuse of quasi-regenerative cycles for the fast simulations in this paper. The two specific rare eventregimes are then investigated in Sections 4 (low load) and 5 (high capacity). In both of theseregimes, the utilization tends to zero. We conclude in Section 6 that these techniques provide asignificant improvement over standard techniques when events are very rare, and indicate scopefor further research.

2 Motivation and Background

2.1 Blocking Probabilities

A cellular network is a collection of � spatially separated base stations, and a collection of userswho make calls of limited duration. During a call, a user communicates with the nearest basestation by means of one of channels. The region which is closer to one base station than to anyother is called a cell. The principle behind cellular networks is that each of the channels canbe used simultaneously by multiple users across the network, if and only if the so-called “reuse

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2

3

4

5

6

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Figure 1: Simple cellular network model

constraints” are satisfied. These constraints ensure that the performance in any given cell is notexcessively degraded by the interference caused by other cells using the same channel. The reuseconstraints hence depend on the precise layout of the cells. For the examples in this paper, weshall assume that the cells form a hexagonal grid and 3-cell reuse is employed; that is, no channelmay be used simultaneously by more than one call in any group of three mutually adjacent cells.In general, a set of cells in which a channel may only be used once is called a “clique”. Figure 1shows a simple seven-cell network, with one clique highlighted. Let � be the number of cliques(in Figure 1, ����� ), and let � � be the � th clique, ����� � � � � � � .

Calls can arrive at the cell in one of two ways. They may be new calls or they may be existingcalls being handed off from neighbouring cells due to user mobility. The model used in this paperdoes not include user mobility. Using dynamic channel assignment, calls arriving to a cell areassigned one of the available channels. If no channel can be allocated without violating a reuseconstraint, then the call is blocked. Otherwise it is accepted, and uses the selected channel. Inpractice, the call will generally use the same channel until it departs from the cell. Thus, in general,the state of the system depends on both the number of calls in each cell (the occupancy), and alsoon which particular channels they use.

There is no useful lower bound on the occupancy of a given cell, � , in the states when blockingoccurs; it is possible for calls arriving to cell � to be blocked when there are no calls at all in cell� , if all the channels are used elsewhere in the cliques to which � belongs. Define the “cluster”associated with cell � to be the union of all cliques containing � :��� ��� ! " � � � �It is then possible to say that the occupancy of the cluster

�#�must be at least $ when calls arriving

to cell � are blocked, since each channel must be blocked by at least one of the cliques containingcell � . That is, %� & ' ( ) �+* $ (1)

is a necessary condition for blocking to occur, where ) � denotes the number of calls (i.e., channelsin use) in cell � . This is the fundamental property of blocking states that is used in the methods

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presented here. Note that when channels are reserved for handovers, as for example in Li and Alfa[2000], then , should be interpreted as the number of channels available to new calls and -/. as thenumber of calls using these channels.

Most of the techniques described in the introduction rely on having a known closed form forthe blocking probability. There is such a closed form for maximum packing channel assignment,proposed by Everitt and Macfadyen [1983], in which channels may be reassigned on the arrivalof a new call. However, the operation of reassigning calls is not feasible in practice, and so thisclosed form is merely a lower bound for the blocking of real channel assignment algorithms. Thetechniques to be presented in Sections 4 and 5 are applicable to real channel assignment algorithmsand are thus of more general applicability than most of the techniques described in the Introduction.

In general, the state of the occupancy process at time 0 is given by 1-32 0 4 562 1-87 9 7 2 0 4 : ; ; ; : 1-/<89 =82 0 4 4 ,where 1-?> 9 @ 2 0 4A5CB if channel D is used in cell E at time 0 , and zero otherwise. Sometime we alsoused a simplified state description, given by -32 0 4F5G2 - 7 2 0 4 : ; ; ; : - < 2 0 4 4 , where - > 2 0 4 representsthe number of channels in use in cell E at time 0 . This is the aggregate occupancy process. Notethat -32 0 4 is completely determined by 1-H2 0 4 . Under maximum packing, all calls in progress canbe rearranged to different channels on the arrival of a call, and so the behaviour of the system isdetermined entirely by this aggregate state.

Some of our numerical examples will use the so-called “clique packing” approximation to max-imum packing, proposed by Everitt and Macfadyen [1983] and further investigated by Raymond[1991], which considers only constraints local to each clique. A state is feasible under cliquepacking if each of the cliques contains no more calls than there are channels:-JI @ K L8M ,ON#P�5�BQ: ; ; ; : R (2)

where - I S L is the number of calls in a set of cells, T , in a given network state, - .At each cell E , new calls arrive following independent Poisson processes with corresponding

intensities U�> : EV5WBQ: ; ; ; : X . A call that arrives at cell E at time 0 is accepted if there is still atleast one channel available. If call rearrangement is not permitted, this requires that there existsa channel, D , such that 1- . 9 @ 2 0 4A5�Y for all cells P[Z]\ > . Under clique packing the requirement issimply that ^V_ `@ K a > 2 -/I @ K L 4 M ,cb]B ;An accepted call on channel D causes 1- > 9 @ 2 0 4F5d1- > 9 @ 2 0 eJ4gfhB (whence - > 2 0 4i5j- > 2 0 e/4gf�B ), allother components of the state remaining unchanged. We say that at this time the call is connected.If an incoming call to cell E finds no channels available (under clique packing, the current statesatisfies (2) with equality for some D .lkmE ) then all channels are used and the call is blocked, withno change to the state. Note that (1) is a necessary condition for blocking whether or not cliquepacking is used.

Calls stay connected for a random length of time called the holding time, assumed to be in-dependent of the rest of the process history. All holding times are identically distributed withmean B n o . When a call in cell E departs, the corresponding occupancy component is decreasedby one unit. Although the holding times are assumed to be exponential in this paper, the networkperformance is in fact independent of the holding time distribution for many channel assignment

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schemes, including clique packing [Kelly, 1979]. Most models without call rearrangement are notincluded in these.

This model gives rise to a continuous-time Markov process. Furthermore, because the process,pq3r s t , consists of independent arrivals and departures, it can be expressed as a quasi birth and death(QBD) process (see Neuts, 1981). In QBD processes, states can be arranged in layers, such thattransitions from layer u can only be to states in layers ulvcw , u or u+x�w . For any y , layer u canbe defined to consist of states in which cluster z#{ contains u calls. This system is a QBD, since acall arrival within the cluster causes a transition from layer u to layer ugx]w , a departure within thecluster causes a transition from u to u|v�w , and an arrival or departure outside the cluster causes atransition entirely within level u . In this representation, all blocking states for cell y are in layers}

and higher. The occupancy process,pq3r s t , is a particular case of a QBD where the rates and

barriers depend on all components of the process. When the process is in statepq , the birth rate of

component y of the aggregate state, q , is ~ { ��� ��Q�� �� � � . Here ��� � � is the indicator function of event �and

p� { is the set of states that cause blocking in cell y , which depends on the channel assignmentused. For clique packing, this simplifies to ~ { ��� �Q�� � � � , where

� { is the set of aggregated blockingstates for cell y : � {J��� q��i�����#� �+� y � qJ� � � � � }F� �Oy8��wQ� � � � � ��� (3)

Here � is the state space consisting of all integer vectors q � r q8� � � � � � q?�|tl�¡  � satisfying (2).When clique packing is not used, define

� { to be all those q���� which are aggregate statescorresponding to the blocking states,

p� { . Note that forpqm� p� { equation (1) holds. The total death

rate at level u is u ¢ .The performance measure of interest is the blocking probability, defined as the long term prob-

ability that an incoming arrival is blocked:£ �C¤ ¥ ¦§ ¨|©¡ª �{ « �#¬ { r s t� r s t � �­ { « � ® ~#{~ tot ¯ £ { � �­ { « � ® ~�{~ tot ¯�° r p� { t (4)

where ¬ { r s t is the total number of calls blocked in cell y up to time s , � r s t is the total numberof arrivals up to time s and ~ tot � ª �{ « � ~�{ is the total arrival rate. The term

£ { is the long termproportion of calls arriving to cell y that are blocked, and ° r p� { t is the stationary probability thatthe state is in the blocking set

p� { .The renewal-reward theorem can be used to re-write (4) in terms of expectations within re-

generative cycles. However, as the state space has many components, regeneration cycles arefrequently too long to be a feasible basis for simulation. The concept of quasi-regeneration wasintroduced to calculate stationary averages for such systems, as explained by Chang, Heidelberger,and Shahabuddin [1995] and Gaivoronski and Messina [1996]. Consider a random process,

pq8r s t ,and assume it starts with the stationary distribution ± r pq3r ²�t �j³ t � ° r ³ t . Consider a set ofstates, � , such that there is an a.s. finite stopping time

p´/µ, defined as the first entry time to the set� from its complement �|¶ . Let

p´ � , p´/· , . . . be the subsequent times of entrances to the set � fromthe set � ¶ . Clearly these are also stopping times, and are a.s. finite. Since the process is in steadystate, the distribution of the process ¸ pq8r s x p´ { tl��sl¹�²�º , y|»�w , is identical to that of the process

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¼#½¾3¿ À3Á ½Â/à ÄAÅ ÀAÆhÇ�È . The set É is called a quasi-regenerative set. Because the aggregated occu-pancy process ¾3¿ À Ä described above is an irreducible Markovian process on a finite state space, allsubsets of the state space are quasi-regenerative sets and a unique stationary measure Ê exists. Thetimes between consecutive entries to the set É are termed “ É -cycles”. Unlike true regenerativecycles, É -cycles may not be independent, but they are still identically distributed.

It will be useful to consider different quasi-regeneration sets, É+Ë , for different cells Ì . FollowingSadowsky [1991], we will require that É Ë8Í ½Î Ë . Let

ÂlÏ Ë Ð be the random length of an É Ë -cycle, andÑ Ë ¿ Â Ï Ë Ð Ä be the amount of time within an É Ë -cycle that the process spends in½Î Ë . Then [Breiman,

1992] Ò ËJÓ�ÔJÕ Ñ Ë ¿ ÂlÏ Ë Ð Ä ÖÔ/Õ Â Ï Ë Ð ÖØ× (5)

The sets É Ë will be chosen in such a way as to minimize the required simulation time. É -cycleshave been used in a number of papers, such as Sadowsky [1991], Nicola et al. [1993] and L’Ecuyerand Champoux [1996], to name but a few.

2.2 Fast Simulation Methods

A commonly used figure of merit of an estimator is its relative efficiency (see, for example, Glynnand Whitt, 1992). This quantifies the tradeoff between computational effort and relative meansquare error (or equivalently, the relative variance if the estimator is unbiased).

Definition 1 Let ÙÚ ¿ Û Ä denote a consistent estimator of

Òthat uses Û samples of a stochastic

process. The relative efficiency of ÙÚ ¿ Û Ä is:Ü�Ý ¿ ÙÚ ¿ Û Ä Ä ÓWÞ Ò�ßCPU Õ ÙÚ ¿ Û Ä Ö à3á â Õ ÙÚ ¿ Û Ä Ö ã]ä (6)

where CPU Õ ÙÚ ¿ Û Ä Ö denotes the expected value of the CPU time of the simulation that produces theÛ samples.

Definition 2 Let¼ ¾3¿ À Ä È be a stochastic process defined on a set of outcomes å and let æ Æ]Ç be a

parameter of the distribution of the process, denoted çJè . The event éëê�å is called a rare event ifì í î è ï à ç è ¿ é Ä Ó Ç .If ð ¿ æ Ä Ó ç è ¿ é Ä is estimated via simulation using ñ#ò ó3ô for Û consecutive É -cycles, then the

variance of the estimator is at least ð ¿ æ Ä ¿ õAö ð ¿ æ Ä Ä ÷ Û if consecutive É -cycles introduce positivecorrelation on the Bernoulli sequence (as is usually the case in our application). The relative errorin the estimation, defined as the standard deviation of the estimator divided by the true value, isthen bounded below by ø ¿ õ|ö ð ¿ æ Ä Ä ÷ ¿ Û ð ¿ æ Ä Ä+ù�ú , as æ ù Ç for fixed Û . The efficiency in theestimation of rare events can be improved using a change of measure approach via importancesampling (see Devetsikiotis and Townsend, 1993 and Asmussen and Nielsen, 1995 among others).

The simulation model assumes an underlying discrete time Markovian process¼ ûHü È , such that

the occupancy process¼?½¾3¿ À Ä È can be determined solely from the generation of

¼ û ü È . Use the

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notation ý#þÿ�� ��� � ���� � �� � � � and ý ÿ�� ��� � ����� � �� � � � for the embedded processes (with theobvious abuse in notation). Let ���� � ��� be the conditional density of ��� ��� given the state þÿ�� ����� þÿ .Let now ����� � � � be another conditional density, and define for any �� "!#�$ � $�% �&� '�( ���� ) � * � ��� ��� �� ��� ) � * � ��� ��� � � (7)

known as the Radon-Nikodym derivative of the distribution + with respect to the distribution + � ,where + and + � are the distributions of the respective Markov processes after � events. If a rareevent , depends only on the history of the process ý ��� � �.- � for some fixed time � , then (see,for example, Ross, 1997) /10 2�3 4�5 6 � / � 0 # $ 2�3 4�5 6 � (8)

where/ � denotes the expectation w.r.t. the distribution induced by � � . The change of measure (8)

is valid also when � is a random stopping time, as explained, for example, in Sadowsky [1991]and Asmussen and Nielsen [1995]. This approach can use arbitrary densities ��� as long as theysatisfy an absolute continuity constraint; the restriction of + to the “important set” , , +�7 4 , must beabsolutely continuous with respect to the new measure + � ; equivalently 8�9: ;, , � �� � � � 9 � �=<:>@?� ��� � � � 9 � �=<:> . (See, for example, Vazquez-Abad and LeQuoc, 2001.)

Definition 3 The unbiased IS estimator for the rare event probability A � B � , #�$ 2 3 4�5 , has boundedrelative error (BRE) under + � if there are constants CED:F � B ( <:> such thatG H�IJ K�J L M N�O P � 0 #�$ 2�3 4�5 6A � B � - C � (9)

The above definition is widely used in rare event estimation; see, for example, Shahabuddin[1994]. The following lemma is a direct consequence of Definition 3. (See, for example, Chang,Heidelberger, and Shahabuddin, 1995.)

Lemma 1 If there are constants Q � � and C such that A � B �ER Q B S and#�$ 2 3 4�5 -���B S a.s., then the

IS estimator for A � B � , #�$ 2�3 4�5 , has BRE.

By construction, A � B �T� + J � , �T� / � 0 #�$ 2 3 4�5 6 under any valid change of measure. Often thenew measure depends on B , and hence, in general,

#�$will be a function of B , although we will not

make this explicit in our notation. Because variances are non negative, it must always be true that/ � 0 #�U$ 2�3 4�5 6 R A U � B � � (10)

Estimators that satisfy (10) with equality are optimal and are called zero variance estimators.

Sadowsky [1991] studies a V@W�X V@W�X Y X�F queueing system and estimates the excessive backlogprobability: the probability that an arrival finds at least Z customers waiting, Z being large. Bydefining the quasi-regenerative set, [ , to be those states in which all Y servers are busy, he finds

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the asymptotically optimal change of measure for the probability of excessive backlog occurringwithin a given \ -cycle. The approach uses the so-called “conjugate” distributions, which in theMarkovian case is known as “rate swapping”: simulate an ]_^ ]:^ ` ^ a queueing system, with thenew rates b�c=d ` e�fge c=dhb ^ ` i (11)

This change of measure is applied to the system once its occupancy reaches ` , and its optimalityrelies on the fact that the number of servers in use is constant between the start of acceleration andthe time when the backlog reaches j . This algorithm can also be used for estimating the fractionof customers lost in the long run in a k@l�^ k@l�^ ` ^ j queueing system, j being large compared to ` .

In Sections 4 and 5 we discuss the implementation of the IS that swaps arrival and service ratesfor the cellular network problem.

3 Model for Fast Simulation

Estimation of blocking probabilities can be especially difficult in the case when blocking is a rareevent. The following sections address fast simulation of blocking probabilities for two regimesunder which blocking is a rare event: low load and high capacity. For both problems we use thesame simulation model, which does not rely on the product form solution.

The occupancy process m�no�p q r s q=t:u�v , as described in Section 2.1, is a continuous time Markovprocess. Recall that the arrival rate of calls into cell w is

b�x, w dzy f i i i f { , and the mean holding

time per call isy ^ e . Also, recall that the blocking probability can be expressed as in (5). Given

a set \ x , consider the process started with the stationary distribution, conditional on the eventm�no"| \ x v . Define } x , as the number of events required to reach a blocking state for cell w or the endof the current quasi-regenerative cycle, and ~ x as the event that the cycle contains a blocking statefor cell w : } x1d:��� � m ������� t_�E� x � or no�p ��� r=| n� x v f and ~ x1d m ��� ��� �E� x � v f (12)

where � � is the epoch of the � th event in the system. Then� x d���� � x p � � x � r � ~ x ��� p ~ x r��� � � x � � i (13)

For each cell w , a change of measure is performed to build an efficient estimator for� p ~ x r . Fix

the index w throughout the remaining sections, until the description in Section 5 of the simulations.We now define the set \ x .Definition 4 Let the quasi-regenerative set n\ x be the set of all states in which the cluster occu-pancy, o � � � � , satisfies o � � � ���_� x for some threshold u�� � x �_j . Let \ x be the corresponding set ofaggregate states.

The optimal choice of

� xwill be considered in Section 5.2.3.

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Acc

eler

ated

pro

cess

ing

Res

me

back

bone

fro

m s

ame

stat

e

Figure 2: “Backbone and ribs” simulation framework.

The following sections will derive changes of measure with bounded relative error for esti-mating ��� �.� � , the probability that there is a blocking state within an  ¡� -cycle associated with anarbitrary cell ¢ of the cellular network model. Fast simulation via IS requires simulating the pro-cess with cluster acceleration only until the process enters a blocking state, i.e., a state in £¤ � . Oncethe state enters £¤ � , the standard measure is used to resume the simulation and to estimate the timespent in blocking states within the current  ¡� -cycle, ¥1¦ §_� ¨E© � ª � « �.� ¬ .

However, as mentioned before, an  ¡� -cycle must start with the state distribution being theequilibrium distribution conditional on the process having just entered the set   � from  ¡­� . Inthe absence of importance sampling, this will also be the distribution at the end of the   � -cycle.However, importance sampling disturbs the distribution, and the distribution of the state at theend of an accelerated  ¡� -cycle cannot be used as the starting state for the next  ¡� -cycle. Hence, inaddition to the simulation with importance sampling mentioned above, we start a second simulationfrom the same initial state (i.e., the initial state that is used for the   � cycle with importancesampling). This simulation is done with the original probability measure and its purpose is torecover the initial state for the simulation of the next  E� -cycle. This gives rise to the “backboneand ribs” arrangement, seen in Figure 2. A second advantage of this approach is that the lengthof an  T� -cycle, the denominator of (13), may be estimated with lower variance from the non-accelerated  T� -cycles. This technique is common in queueing network simulation, and has beenimplemented by Nicola et al. [1993], Chang, Heidelberger, and Shahabuddin [1995] and L’Ecuyerand Champoux [1996], among others.

For each cell ¢ , the  ¡� -cycle is used to get estimates only of the blocking probability ®E� ina single cell. Thus in order to estimate the blocking probability, it is necessary to run separatesimulations for each cell. Fortunately, it is not necessary to simulate the “backbone” separately.Instead, separate “ribs” can be started each time the backbone starts an   � -cycle for any cell ¢ .Note that a single event may be the start of  ¡� -cycles for more than one ¢ , and each of these mustbe considered.

The ribs may be further simplified by noting that the set   ­� contains no blocking states for cell

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¯. Thus, once an accelerated °¡± -cycle has left the set °¡± , the entire time spent in blocking states

will already have occurred. This means that the rest of the ° ± -cycle need not be simulated, eventhough a considerable amount of time may be spent in the set °E²± before the ° ± -cycle is truly over.

The simulation model is the standard clock technique of Vakili [1991], which corresponds tothe dynamical description of a multidimensional birth and death process. Consider the embeddedoccupancy processes ³�´µ�¶ ·�¸ ¹ and ³ µ�¶ ·�¸ ¹ . Let º ¶ ·�¸¡»½¼�¾¿ À�Á µ ¿ ¶ ·�¸ be the total occupancy at step· . An exponential random variable with intensity ÂTÃ�Ä Å Æ is used to determine the inter-event time,Ç Å È Á , or the time until the next event. Here

 Ã�Ä Å Æ » ¾É ¿ À�Á1Ê ¿�Ë º ¶ ·�¸ Ì�ÍNext, the event type Î Å È Á is determined as a discrete random variable with distribution

Î Å È Á » ÏÐÐÑ ÐÐÒ=Ó ¿¡Ô arrival at cell Õ w.p. Ê ¿Â Ã�Ä Å ÆÖ ¿TÔ departure of call at cell Õ w.p.µ ¿ ¶ ·�¸ ÌÂ=Ã�Ä Å Æ Í

The embedded occupancy process is updated by first settingµ ¿ ¶ · Ë:× ¸�» ÏÑ Ò µ ¿ ¶ ·�¸ Ë:× if Î Å È Á » Ó ¿ and ´µ�¶ ·�¸¡ØÙ ´Ú ± ,µ ¿ ¶ ·�¸ if ÎÛÅ È Á » Ó ¿ and ´µ�¶ ·�¸ Ù ´Ú ± ,µ ¿ ¶ ·�¸�Ü × if ÎÛÅ È Á » Ö ¿and then determining the state, ´µ�¶ · Ëh× ¸ , using the channel assignment rule. Recall that Ý Å is theepoch of the · th event in the system.

When call rearrangement is permitted, ´µ�¶ ·�¸EØÙ ´Ú ± if and only if µ�¶ ·�¸EØÙ Ú ± , and so the evolutionof the aggregate process is determined only by its state. When calls cannot be rearranged, therandom variables Î Å and the inter-event times are as described above, but an additional decisionis made concerning which channel an arrival is to, or a departure is from. These decisions affect´µ�¶ · Ë:× ¸ but not µ�¶ · Ë:× ¸ . Without rearrangement, the decision whether or not to block an arrivalcannot be made on the basis of µ�¶ ·�¸ alone, but µ�¶ ·�¸ must still satisfy (1) for blocking to occur.

To denote use of IS for the standard clock simulation model we use the acronym ISSC. Similarwork has been done in the context of reliability by Heidelberger, Shahabuddin, and Nicola [1994].

4 Static ISSC Estimation for Light Traffic

In the light traffic regime, assume that Ê ± »h· ± Þ ß ¯ » × ß Í Í Í à , with ÞTáãâ . (This is analogous to theregime used by, for example, Shahabuddin [1994] in the context of reliability.) In the ä@å�æ ä�æ�ç æ ècase, the servers are busy from the start of the ° -cycle containing a blocking state, until a blockingstate is reached. In our model, however, channels are not continuously busy within ° ± -cycles untila blocking state is reached. Thus swapping rates as in (11) will not be optimal. Instead, consider

11

the change of measure that swaps aggregate arrival rates per cluster and inverse expected holdingtimes.

Proposition 1 Consider the ISSC simulation model with initial state éê�ë ì�í such that ê�î ï ð ñ ë ì�í.òó ô òõì , and ö ô -cycles as defined in Definition 4. Arrivals at the cluster ÷ ô have rate ø�ù ò½ú andservice rate for the calls in the cluster is ú ù ò ø . Other inter-arrival and holding times (outsidethe cluster) have the original exponential distribution. Call the underlying measure û ù . Then

û ë ü ô í�ò�ý ù@þ ÿ ��� ��� ë ú � ø í�� ð �� � ë ê î ï ð ñ ë ��í ��� í � � � ��� øú���� ������� � ð � "!where ê1î ï ð ñ ë ��íÛò$#�% & ï ð ê % ë ��í is the total occupancy of the cluster, ' ë (�í is the total number ofarrivals to cell ( prior to event number ) ô (including blocked calls), ' ò #�% & ï ð ' ë (�í and * is thecorresponding number of call completions (excluding blocked calls).

Proof : Given the state ê�ë ��í of the process at the time of the � -th event, � � � ,+ ÿ �� ë - ù. î � ñ í . Thenew event rate is the random variable- ù. î � ñ òhú"/ �% 0& ï ð ø % / ø �% & ï ð ê % ë ��í�/ ú �% 0& ï ð ê % ë ��í !and the event types are now

1 � � ò2333333333334 333333333335

arrival at cell ("6 ÷ ô w.p.ø ôø ú- ù. î � ñ

arrival at cell (876 ÷ ô w.p.ø %- ù. î � ñ

departure of call at cell ("6 ÷ ô w.p.ê % ë ��í ø- ù. î � ñ

departure of call at cell (876 ÷ ô w.p.ê % ë ��í ú- ù. î � ñ

9On ü ô , we perform this change of measure within an ö ô -cycle until the ) ô -th event occurs,

which is the first time that the state is in é: ô . Using the framework of Section 2.2, the correspondingRadon-Nikodym derivative is given by (7) as; � ð ò � ð �<� � - . î

� ñ- ù. î � ñ �>= î ?@AB C D ��? AEB C D ñ F C G�H (14)I � ð �<� � - ù. î� ñ- . î � ñ � � �% & ï ð � ø %ú�ë ø % J ø í � ��� K C G H � L � /NM ú øPO ��� K C G�H & Q ð � / ��� K C G�H 0& Q ð R S�ð � �

12

where TVU�W�X Y Z"[�\^]>_�U ` is the set of event types which are arrivals to the cluster and similarlya U WbX c Z [�\d]e_ U ` is the set of event types which are departures of calls within the cluster. Fromtheir definitions, it follows thatf�gh�i j kml>f h�i j k Won p lrqEs�t n qVl p sEuZ v w x y Z n z s W l n p lrqms n y i w x k n z s�l�{ s |Simplifying the expression above,}�~ x W�� ����� l n p l>qms ~ x ��u j � � n y i w x k n z s�l�{ s � j � � ��� �Z v w x�� qp,��� i Z k �e� p qP���W�� ��� � l n p l>qms ~ x ��u j � � n y i w x k n z s�l�{ s � j � � � � qp ��� � � | (15)

Application of (8) proves the claim. �Lemma 2 When

q�� p , }P~ x ��� ��x � � n qE� p s � �� x  ¢¡ g�lw.p.1. (16)

In particular,}P~ x ��� ��x � �£{   ¡ g l w.p.1., guaranteeing variance reduction when using the ISSC

estimator.

Proof : On ¤ U , there are at least ¥ on-going calls within the cluster at the hitting time ¦ U . Atthe time the §�U -cycle begins, there are (by Definition 4) ¨ U t©{ calls within the cluster. Becausethe stopping time within an § U -cycle counts only the transitions from the start of the § U -cycle upuntil a blocking state is reached, and ª« U�¬ ª§ U , it follows that on the set ¤ U we have y i w x k n z s"­¨ U t�{   z"W {   | | |   ¦ U , whence n p lrqEs ~ x ��u j � � n y i w x k n z s�l�{ s � j � � ­¯®�| (17)

Moreover, Y"W©° Z v w x YEn \ s�­ c t ¥ l ¨ U . Using (15) gives the result. �Theorem 1 The ISSC estimator for ±�n ² s , suggested by Proposition 1, has BRE as ²�³ ® when¨ U W ® and

q Z W¯z Z ² , for all cells \ .Proof : The proof is an application of Lemma 1. The upper bound is obtained with the result ofLemma 2: }�~ x � � � x �,´ � qp,� � W¯µE² �   (18)

13

where ¶�·£¸ ¹�º » ¼ ½ ¾ º ¿ ÀmÁm .

It remains to show Ã�Ä Å Æ,Ç¯È Å Â . In order for blocking of arrivals to cell É to occur, it is sufficientfor the occupancy of a single clique, Ê ºdË É , to reach Ì . Let a “minimal path” be a trajectory inwhich the first Ì events in an Í�Î -cycle are arrivals to the same clique within the accelerated cluster.All minimal paths will lead to blocking states, and thus their probability is a lower bound for Ã�Ä Å Æ .The probability of such minimal paths is the probability that each of the first Ì events be an arrivalto the same clique, and so Ã�Ä Å Æ,Ç�ÏÑÐÒ ÎÓ,ÔPÕ ÂEÖ × Â Ç�ÏØо Î ÅÌ ÀdÙ Ò tot × Â ·¯È Å ÂPÚ (19)

where ÐÒ Î is the smallest aggregate clique rate within cluster Û�Î and È�·NÄ Ð¾ Î ¿ Ä Ì ÀdÙ Ò tot Æ Æ Â , with о Îthe smallest of ¹¯Ü » Ý Þm¾ Ü over the cliques Ê º�ß Û Î .It follows by Lemma 1 that in this case, ISSC has BRE. à

Note that the condition á Î�·¯â is not a major limitation, since the modal cluster occupancy will bezero, and so á Îm·¯â gives the shortest Í�Î -cycles, which is desirable for simulations.

The ISSC with á Îd·ãâ and Ìä·æå was tested for Å�çèâ on three sizes of network: 4, 7and 37 cells. The load in Erlangs per cell is given by

Ò Î ¿ À , which was equal for all cells. A singlebackbone was used to estimate all é�Î . The first ê â ë ribs were simulated for each É , and used toestimate the numerator of (13). The variance corresponding to each é�Î was estimated using batchmeans, with 100 batches of ê â�ì�ÍíÎ -cycles. The estimation of this variance, as well as that of thevariance of the estimator of é , is described in more detail in Section 5.2.2.

Figure 3 compares the results for clique packing with those of the filtered Gibbs sampler ofVazquez-Abad and Andrew [2000]. The results clearly show that the relative error of the estimatednetwork blocking probability is bounded as Åíçãâ .

Figure 4 shows the performance of ISSC for a seven cell network when existing calls cannot berearranged. In this case, the network has no product form solution. Channels were assigned usingthe first fit algorithm, which starts searching from channel 1 and selects the first available channel.This was shown by Yates [1997] to produce significantly less blocking than random selection.

Along with the “non-accelerated Í�Î -cycles”, Figure 4 includes results for a “simple” estimationscheme. This simulates only the backbone and simply counts the proportions of calls which areblocked. For high blocking, this has higher relative efficiency than the simulation based on quasi-regeneration. Its CPU time is lower partly due to its ability to estimate blocking over the entirenetwork at once, rather than focusing on a single cell, É , in each Í�Î -cycle, and partly due to theelimination of “bookkeeping” associated with tracking the Í Î -cycles. However, for lower blockingit performs even worse than the non-accelerated Í Î -cycles, because it does not keep track of theproportion of time that the network is in a blocking state, but simply the number of calls blocked.

14

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Figure 3: Relative efficiency and relative error for importance sampling (IS) and filtered Gibbssampler (FGS) for light loads

0.0001

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Figure 4: Relative efficiency for light loads when ribs are accelerated, when ribs not accelerated,and when the simple algorithm is used.

15

5 Dynamic ISSC Estimation for High Capacity

5.1 Change of Measure

It is unlikely that future networks will be operated at extremely low load, as was assumed inthe previous section. Engineers are more interested in the behaviour as the capacity increases.This is particularly true of wavelength-continuous wavelength division multiplexing (WDM) trunknetworks, which are mathematically analogous to the cellular networks described so far. It ispossible to overcome the restriction that îNïñð used in Lemma 2, and to investigate limitingregimes other than îmò ð^óæô , by allowing the arrival and service rates to be state dependent in thenew probability measure.

This section addresses the simulation of the regime of õöóø÷ with îù ò ð independent of õ .Swapping î and ð as in the previous section will not yield BRE for high capacity regimes (evenwhen î8ï�ð ) because ú�û ü ý ù þ óãô as õbóæ÷ . A change of measure, which is optimal in the caseof a single clique cellular system with õ channels, is presented and then applied to the generalcellular case. Notice that the former is equivalent to a queueing system with õ parallel serversand no waiting, and can be solved trivially using the Erlang loss formula. However, the changeof measure and corresponding simulation analysis are presented to give the intuition for moregeneral cellular models. In the general cellular context it is suboptimal, but provides a dramaticimprovement over simulation using the original measure.

5.1.1 The single-clique case

In the case of a single-clique network, whether ÿ� ü � þ�� ÿ� ù or not, is completely determined by theaggregate state � ü � þ , so we may just use the process � � ü � þ � . Note that � � ü � þ � is a birth and deathprocess on ��� ô�� � õ � with birth rate î and death rate � ð , for � � � , and call � � ü � þ � theembedded random walk (with the obvious abuse of notation). Let the quasi-regenerative set be� �� ������� � õ � , and let � ��� be the random length of an

�-cycle. Let � ��� be the first

hitting time (in the embedded random walk) of õ within the�

-cycle; that is, ����� � !"� �$#&%('�)*�or � ü � þ �bõ � , where % ' is the epoch of the � -th event. Finally let ý+�,� %"-�ï.� � . Consider adynamic change of measure, where the birth and death process � û ü � þ has rates î û ü � þ and � ð û ü � þ .Theorem 2 Consider the ISSC estimator for úPü ý þ using the dynamic ratesî û ü � þ � î/�0��ü ð�1>ð û ü � þ þ � (20a)ð û ü �2�3� þ � îEðî û ü � þ (20b)

for �4)3�5��� , starting with ð û ü �2��� þ ��ô , and ð û ü � þ �¯ð , î û ü � þ �¯î for ��ï3�5��� . This has BREin the limit of õbóã÷ , with the likelihood ratio6 - � 798�:;< =�> ? : îî/�0��ü ð�1>ð û ü � þ þ (21)

when a blocking state is reached. Moreover, the variance of the estimate is zero even for finite õ .

16

Proof : With @BA C D�E/F GIH�J , K A C LMG2HNF since the O -cycle is not allowed to end until a blocking stateis reached. This violates the usual absolute continuity condition that for every P0QMR , K2C P5G2S3J4TK A C P5G5S3J . However, as mentioned in Section 4, K"U V is absolutely continuous with respect to K A .This follows because on the event L , a blocking state is reached before the O -cycle is over, thusno trajectory on L can have a transition from D�E�F back to D before W , as that would start a newO -cycle. Thus for every P*QXL , KIC P5G�S*J TYK A C P5G�S*J , and the change of measure is valid forthe estimation of KIC LZG .

On any path leading to the blocking boundary, [ , any transition due to a call departure fromstate \ ( \�]D/EN^ ) to \4_.F must necessarily be followed in some future stage by a matchingtransition from \5_�F to \ ; otherwise it is impossible to achieve full occupancy. The correspondingfactors contributing to `Ia are thenb \ @\ @ A C \ G c bedd A C \f_3F G c H b2d A C \5_3F Gd c bgdd A C \f_3F G c H*F�htherefore all such loops cancel out their contributions. The only remaining contributions to ` aare the factors for the “minimal” blocking path D�E3F�ijD�E0^�ikD�E�l0inm m m�ij[ , whichyields (21). This is a deterministic function of [ . Since K A C LZGfH.F , `Ia&o&p VIqfHr9s `Ia&o&p VIq t w.p.1,and is thus optimal. u

Note that since this change of measure is exactly optimal for any combination ofd

, @ and[ , it is optimal for any scaling regime. This includes the two scaling regimes considered in thispaper, and also the important regime where [*iwv with

d9x [4@ held fixed, which is not explicitlyaddressed here.

Note also that for a fixed @ A C D�E�F G , the rates are independent of [ . The optimal adaptivity to[ comes from the fact that the rates change as the actual current occupancy changes.Asymptotically, the change of measure under the new rates is analogous to the static change

of measure for an y x y x zMx v queue that swaps arrival and service rate — as in (11), as the nextLemma shows. Note that acceleration of an y x y x zMx v queue can be static, since the numberof active servers remains constant as the occupancy tends to infinity, leading to a constant servicerate during the acceleration. Rate swapping thus gives a static change of measure. In contrast, thenumber of active servers in the system considered here is equal to the instantaneous occupancy,and the total service rate changes throughout the acceleration. The change of measure of (20) isdynamic, reflecting this change.

Lemma 3 Under the update rule of Theorem 2, for any initial J/{�@(A C D�E3F G2|3@ ,} ~ �� �f� d A C \ G x \wH�@} ~ �� �f� @ A C \ G \wH d mProof : We will first show that @BA C \ G$ijJ as \Zikv . By induction, J0{@BA C \ G�|@ for all\4]3D�E3F , and hence � @BA C \ G � has a convergent subsequence. To see that @ is not an accumulationpoint, note that this would imply @BA C \ GIH�@4_Z�(C \ G for some �(C \ G2H��&C F G , �(C \ G��� J . By (20b), that

17

would in turn imply ��� � � �B�0���N�B� � � , but by (20a), �"� � � �(�0���N� �(� � � , which is a contradiction.Thus there is a strictly increasing sequence ��� ���f�X� and a �B���0� ��� �B� such that �9� � ��� �M� �5���9�as �,��� . For any such sequence ��� �M� , there is a ��������� �f� � such that� � � ��� � �3¡ � �(�0� � � ��� �M� �+�¢� ��� � �3¡ �(�0��� �M� � � ����� � �B��£�� ¡ �¤ �f��£�� ¡ � ¥Thus �"� � ��� ��� �I�w� , and by (20b), �B� � ��� �M� �2�¦�B�5��� . Since the sequence �&� �M� was arbitrary,0 is the unique accumulation point, and �B� � � �2�¦� as ����� . From (20a),��� � � �� �N� �$��� � � � � �9� � � �¦�(�and the result follows from (20b). §5.1.2 The general cellular network case

Consider again the standard clock simulation model of the cellular network, and let the changeof measure for estimating ¨I� ©�ª � be such that the total event rate is the same as for the originalmeasure: « �¬ ��« ¬ , when the total occupancy is ­ . When an event occurs, it is an arrival to (or adeparture from) cell ®X¯�M° ª with probability �&± ² « ¬ (respectively, ³�± �B² « ¬ ) just as for the originalmeasure. Arrivals to (departures from) the cluster ° ª will now occur with probability �"� � � � ² « ¬(respectively, � �9� � � � ² « ¬ ). The proportion of arrivals to the cluster that go to cell ®M�X°�ª remainsfixed at � ± ²�� . Let ��� ´���� ³Bµ ¶ · ¸ � ´�� be the embedded random walk, under the new distribution ofthe process, starting with ´ �*¡ as the start of the ¹�ª -cycle.

In the standard clock simulation, �B� � ��� ´�� � and �"� � ��� ´�� � determined the event type, where �B� � º �and �"� � º � satisfy the recurrence relation (20), starting from �B� � » ª ��¡ �2��� (or more generally from��¼*�B� � » ª"��¡ �f½*� ). Under this change of measure, the rates are no longer constant, but dependon the state (more specifically, on the cluster occupancy), hence the name “dynamic ISSC”.

It is straightforward to calculate the Radon-Nikodym derivative¾I¿ · � ¿ · À"ÁÂÃ Ä Á Å�Å �� � � ��� ´�� � Æ�Ç&È É"Ê Ë Ì Í Î · Ï � Å �� � � ��� ´�� � Æ�Ç&È É9Ê Ë�Ì Í Ð · Ï � Ç&È É9Ê Ë�Ì ÑÍ Ð · Ò Î · Ï Æ �independent of the inter-arrival times, where Ó ª and Ô4ª are as in (14).

In the cellular network case, it remains true that to reach a state Õ³I� Ö ª �X� Õ× ª all backwardtransitions in the cluster °&ª from � to �(��¡ will cancel out forward transitions from �(��¡ to � . Thisfollows from the observation that the cluster occupancy itself can only increase or decrease by 1 ateach event that changes its occupancy. However, it is possible that an arrival to a cell ® �M° ª , ®M¯��Ø ,will be blocked even when Õ³I� ´��/¯� Õ× ª , if Õ³I� ´���� Õ× ± . Since these events do not cause a change inthe occupancy, � , their effect is not cancelled out by departure events. However, the contributionof these events to

¾2¿ · is �"²�� � � � �5½�¡ , and they cannot cause an increase in variance with respect to

18

the original measure. Moreover, these events become less frequent in the rare event scenario, sinceÙ9Údoes not focus on ÛÜ(Ý .Hence the only transitions that will contribute to the final expression for Þ5ß à are blocked ar-

rivals, and the forward transitions from á â2ã.ä to the full occupancy å(æ ç à è é ê â ë . Now the ISSCestimator is Þ ß àBì3í î ï à ð æ ß à è ñ�òóô õ�ö à ÷ òwø ùù ã0ú é û�ü�û Ú é ú ë ë ý

ò ÷"þ æ ô è&ÿwhere � é ú ë is the number of blocked arrivals to cluster �"â while it is in state ú . The final clusteroccupancy å æ ç à è é ê â ë satisfies ���*å æ ç à è é ê â ë������ , where � is the number of cliques in � â , whichdepends on the interconnectivity of the network. The variance of Þ ß à is thus dependent on thevariation of the distribution of the cluster occupancy when a blocking state is first reached.

5.2 Implementation Considerations

5.2.1 Subsampling the ribs

The correlation between consecutive fâ -cycles in the backbone can be very significant. In order toreduce this, it is possible to subsample the ribs, i.e., start a rib for every th �â -cycle in the back-bone. This greatly increases the amount of work required to simulate the backbone. However, anfâ -cycle which is blocked may be very much longer than a “typical” �â -cycle, especially when theload is a small fraction of the number of channels, and so the backbone is often a small proportionof the simulation time. Moreover, the backbone is shared between many cells, making subsam-pling very worth while. This approach is valid for any , but the following heuristic arguments canbe used to select a value to increase the efficiency.

Estimating the variance of a ratio can be performed following Alexopoulos and Seila [1998],even when both numerator and denominator are sample averages of Markov processes with expo-nentially decaying covariances, instead of iid random variables. Let� � ì��� � ��� � ��� � � õ�� � ü�� â � �be a random variable obtained during the � -th �â -cycle. Here � � represents a sample of � é � æ â è ëand � � represents a sample of �4æ â è . The scaling of � � by cancels the subsampling by , so that "! # �� õ ò � � $ ì&% . Then the estimator obtained with ' consecutive �â -cycles satisfies(*) + !�,� â é '2ë $ ì (*) +.- �/ # /� õ ò � � � � � �0� � � õ�� �ò/ # /� õ ò � �2143�5 ò ( ) +.-9ä' /6 � õ ò � � 1 ÿwhere 5 ò is a constant depending on

�! é � æ â è ë 7 $ , but independent of the sample size ' . Recall that� � is estimated from the backbone, while � � is estimated from an independent simulation of a rib.Neglecting the resulting cross-correlation gives(*) +�- ä' /6 � õ ò � � 143�598 é �:�'2ë9ã 59; :�' ÿ (22)

19

where constants <9= and <9> depend on the autocovariances of sequences ? @�A B CA D�E and ? FHG I0A B CA DJErespectively, but are independent of K . In particular, if the sequences ? @ A B and ? I A B have noautocorrelation, then < =�L�M*N�O P @ A Q and < >4L�M*N O P F G I A Q . These constants can be estimated froma single simulation by estimating the variance (22) using different rates of subsampling, R E and R�S ;the difference between these estimates is approximately < =UT R S.V&R E W X K . Note that < = dependson R unless ? @"A B is uncorrelated. Thus (22) should be seen as a linearisation, and the values of R�Eand R�S chosen accordingly.

Also,CPU P�YF G T K W Q�L�Z = K X R\[ Z > K*]

where Z = and Z > are the mean lengths of ^ G -cycles corresponding to the ribs and the backbone,respectively. The relative efficiency (6) is then maximized by settingR�_�`ba Z =<c= < >Z >ed (23)

These values can be estimated coarsely from a pilot simulation, which can also serve as the “warm-up” to achieve steady state. When the accelerated simulation algorithm is used (i.e., the backboneis not accelerated but the ribs are), R _ is of the order of 10, to within one order of magnitude.When the non-accelerated simulation algorithm is used (i.e., when neither the backbone nor ribsare accelerated), R _ is actually often less than 1, indicating that there may be value in runningmultiple ribs from the same point in the backbone. For the numerical results in this paper, R wasnot selected from (23). For cases where the ribs were expected to be short, including all cases inSection 4, R Lgf was used. For more difficult cases, R Lgf h was used. These values were selectedfrom prior simulations.

5.2.2 Variance estimation

The above expressions for variance are very approximate, and are only appropriate for determiningsuitable subsampling rates. The variance of the estimator, YF G , was determined using the methodsdescribed in Alexopoulos and Seila [1998]. This uses batch means to determine the variancesand covariance of the estimators for the numerator and denominator of (13) which can be usedto derive the variance of YF G . This in turn can be used to construct a confidence interval for F G .Since the estimates for the individual F G s are derived from the same backbone, the variance ofthe overall estimator, YF , also contains some covariance terms. These terms are generally expectedto be small because the main source of variance is estimating the numerator of (13), and the ribsfor different cells are simulated independently, albeit with dependent initial states. Hence thesecovariance terms may be neglected, as they are in the numerical results presented in this paper. Ifprecise confidence intervals are required, then one option is to estimate the F G ’s using independentbackbones, with the obvious substantial reduction in efficiency.

5.2.3 Choice of quasi-regenerative cycles

When the load, i X j , is not negligible, the probability that a cluster will be completely empty issmall. Thus if k is too small, like k Llh as is used for single server queues, then the ^UG -cycles

20

become unmanageably long. This has several implications. The most obvious result is that thesimulation time increases in proportion. The seriousness of this is to some extent alleviated bythe fact that longer m�n -cycles produce better estimates of the proportion of time spent in blockingstates within an mHn -cycle.

The more serious problem with long m�n -cycles is that the blocking states become a small pro-portion of the m n -cycles, even given that blocking occurs. The assumption behind the IS schemeproposed here is that mHn -cycles which contain blocking states are rare events, but if an mHn -cycledoes contain blocking states, they form a significant proportion of it. Thus the system is acceler-ated until the first blocking state is reached, and is then allowed to relax back to finish its mUn -cycle.If oJp q and r are both large, then empty clusters become rarer than blocking, and most mUn -cyclescontain a period in blocking states, reducing the effectiveness of the acceleration.

The length of m�n -cycles is minimized by maximizing the rate of crossing the boundary betweensets m n and m�sn of the embedded Markov chain. Note that the stationary rate at which the processcrosses from any state with t0u v w x0y�z n to t0u v w x*y&z n {}| equals the stationary rate at which it crossesfrom t u v w x y�z nJ{&| to t u v w x y�z n . This rate is state dependent:o�~�� �t�� ���t}� �� � z n � �where �� � z n � is the subset of the state space where t u v w x y�z n . This is maximized at the modeof the stationary distribution ~�� � � , which is z nc���&� � v w o � p q , assuming that blocking does notsignificantly distort the state distribution, and that the mean and mode of ~ are close.

Next, in order to guarantee that mHn*� �� n , and hence that � � w � | whenever blocking occurs, itwas required that z n*�&r . Thus we use the value

z n0y&�"� �}�Jr��&| � �� � v w o � p q��e� (24)

There may also be benefit in using values of z n larger than �e� � v w o � p q (but less than r ). Itmeans less simulation for the ribs, but longer mHn -cycles in the backbone. However, the mHn -cyclesin the backbone are shared between all cells.

5.3 Simulation Results

The dynamic change of measure was shown to have BRE for the probability of blocking statesoccurring within an m�n -cycle, � � ��n � , in the case of a single cell (or more generally a single clique).However, (13) shows that the efficiency of estimating the blocking probability also depends on theefficiency of estimating ���  "n � ¡ u n x � ¢ �"n £ and ��� ¡ u n x £ . Figure 5 shows the relative efficiency for theactual blocking probability in the single cell case, for the accelerated and non-accelerated methods.Here ¤cy�o n p q is the load in Erlangs. Again 100 batches of | ¥�¦*m n -cycles were used. Subsamplingwas by a factor of §"y�| (no subsampling). As r increases, the proportion of time in each mUn -cyclespent in a blocking state decays, even on those mUn -cycles which contain blocking. This accountsfor the slight reduction in efficiency as the blocking rate decreases. However, this reduction is verymuch smaller than that which occurs without acceleration.

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effic

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accnon-accρ = 0.5

ρ = 1ρ = 2ρ = 4ρ = 8

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0.1

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accnon-accρ = 0.5

ρ = 1ρ = 2ρ = 4ρ = 8

(b) relative efficiency against blocking

Figure 5: Relative efficiency for importance sampling (ISSC) and ¨ -cycle framework without IS,both in a single cell network.

Figure 6 shows the relative efficiency of the accelerated and non-accelerated methods for aseven-cell clique packing system, for a range of loads, ©gª¬«�­ ® ¯ , and a range of normalizedcapacities, °\® © . The load on each cell was the same.

Again 100 batches of ± ²�³*¨�­ -cycles were simulated from a backbone shared by all cells, ´ . TheISSC subsamples the ¨�­ -cycles in the backbone by a factor of µ}ª¶± ² . The simulations withoutacceleration use µ�ª·± , as the rib ¨ ­ -cycles are shorter and the variance of the correspondingestimator is higher. These results do not suggest that ISSC has BRE for network blocking as°�¸º¹ . However, IS substantially reduces the rate at which the performance degrades for large° .

The reason for the reduced efficiency is that the acceleration is applied to all cells in a cluster.For a constant load, as ° increases the (true) expected cluster occupancy on blocking satisfies»0¼ ½0¾ ¿ À Á  ÃÄ ­ Å ® °¶¸Æ± , since arrivals at each cell are independent, and only one clique need be full.However, because the acceleration is applied to all cells in the cluster, the cells outside the cliquewhich caused blocking are also filled up. Thus the expected cluster occupancy at blocking underthe new measure is significantly larger than under the original measure. That is, outcomes with ahigh cluster occupancy are accelerated too much, thus increasing the variance.

It seems from Figure 6(a) that the improvement decreases as the load increases. However, thisis largely due to the fact that the rate at which the blocking decreases for increasing ° is different.Figure 6(b) shows the relative efficiency against the blocking probability. This shows that thechange in slope of the curves is similar over a range of loads.

Figure 6(b) also uses the “simple” estimator used in Figure 4 which merely counts blockedcalls. In this case, this shows an approximately constant improvement by a factor of around 10compared with the non-accelerated ¨H­ -cycles.

Figure 7 shows the results for the same simulation parameters as Figure 6, but for a seven cell

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0.0001

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r = 1r = 2r = 4r = 8

(a) relative efficiency against normalized capacity

0.0001

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accsimpler = 0.5

r = 1r = 2r = 4r = 8

(b) relative efficiency against blocking

Figure 6: Relative efficiency for importance sampling (ISSC), ÇUÈ -cycle framework without IS, andthe simple simulation, all in a 7 cell network with clique packing.

network when existing calls are not rearranged and first-fit channel assignment is used. Note thatin the range which is of most interest to engineers, with blocking between É Ê�Ë�Ì and É Ê�Ë�Í , theacceleration consistently outperforms the non-accelerated simulation.

6 Concluding Remarks

This paper has addressed fast simulation for estimating blocking probabilities in cellular networks.Blocking is a rare event when the load is low, or the number of available channels is high. Weimplemented the main ideas of fast simulation using the standard clock framework for simulation.In the case of low load, the proposed change of measure yields an estimator that has boundedrelative error. For high capacity systems we propose a change of measure that yields a zero varianceestimator for the single clique case; we were unable to prove any efficiency results with this changeof measure (in the rare event setting) for the general network case. Nonetheless, this change ofmeasure provides significant improvements over standard simulation in more general networkswhen events are rare. The reason for the suboptimality is that trajectories with large numbers ofcalls in a cluster get accelerated disproportionately.

There is much scope for improvement of this technique. The performance for relatively highblocking probability ( Î¶É Ê Ë�Ï ) is poor because of the variable number of calls in a cluster whenblocking first occurs. This may be improved by reducing the acceleration applied to cells in thecluster which are not in the fullest clique. Also, the current need to use separate ÇUÈ -cycles toestimate the blocking probability of each cell limits the scalability of the technique. It will alsobe important to expand the technique to other performance measures and more general systemmodels, such as determining the probability of dropping due to blocked handovers in a system

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accsimple

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(b) relative efficiency against blocking

Figure 7: Relative efficiency for importance sampling (ISSC), ÐUÑ -cycle framework without IS, andthe simple simulation, all in a 7 cell network without call rearrangement.

incorporating user mobility.

Acknowledgement

The authors would like to thank the guest editors and referees, especially Perwez Shahabuddin,who thoroughly reviewed early versions of this paper. His insightful suggestions helped us tocreate a much improved contribution.

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